On finite T-groups and the Wielandt subgroup

The of a group is the intersection of the normalizers of all the subnormal subgroups of . A is a group in which all the subnormal subgroups are normal, or, equivalently, a group coinciding with its Wielandt subgroup. We investigate the Wielandt subgroup of finite solvable groups and, in particular, find new properties and characterizations (see Theorems 1, 2 and Corollaries 4, 6) for this subgroup in the case that is metanilpotent. Furthermore, we provide new characterizations for finite solvable -groups in Theorem 7.